My heater will not get warm - 2001 Ford Taurus

Get the engine and coolant to operating temperature. The upper hose to the radiator should NOT get warm over time, but should get HOT when the thermostat opens. If it warms, then it is leaking and allowing coolant to bypass, and you won't get full heat.
Once warm, check BOTH hoses to the heater core, on the passenger side, the 5/8-3/4" diameter pair. BOTH should get too hot to keep touching. If only one is hot or warm, your heater core is likely clogged or plugged, and restricting coolant flow. Flushing may help, and if not, replacement should cure that problem. tom

Either your heater blend door is failing, your core is plugged, or you have poor coolant circulation. checking the temp of both heater hoses will rule out the core usually. (Warm engine, heater ON, both hoses should be HOT and near same temperature). If 1 hose is hot and the other cold, your core may be plugged. If both hoses are HOT, and still no heat inside, your heater box is likely not directing the warm air properly on the interior. If neither hose is hot or warm, and heater is ON, your heater control valve should be checked to make sure it is flowing coolant to the heater.

If all THAT seems to pass (heater valve definitely open) an d still low or not heat inside and both hoses the same, your water pump may not be making enough flow to feed the engine and the heater. Typically if this is the problem, holding the engineat high rev (3000+) for 5-10 minutes will cause the heater to warm up mostly, and then dropping the RPM to 1000 or 1500 you will feel it cool off rapidly. If that is the case, you will need cooling system repairs.

Most likely the thermostat broke. I expect that you have an older Hyundai and never had the anti-freeze changed. When the engine is cold the thermostat is closed which causes the engine to warm up more quickly. A warm engine runs more efficiently (less gas) and more smoothly. When the engine temp reaches a certain temp the thermostat opens and then allows the hot engine coolant to run through the lines to the heater core. The heater blows air over the heater core which produces the warm air that warms up the cabin. Thermostats used to be cheap and easy to change. I would suggest a web search for how to replace the thermostat. Something like 'hyundai accent 2002 replace thermostat".

Either your thermostat is broken and jammed open (this means the water in the heater matrix will take forever to get hot) or you have an air-lock in the heater matrix. To check the thermostat, the engine should get too hot to touch before the radiator gets even warm. If they all get warm together, your thermostat is faulty. To bleed the heater matrix, uncouple the higher of the two hoses from the bulkhead an top up the water resevoir until you get flow from the loosened heater hose.

UNHOOK BOTH HEATER HOSES AND FLUSH OUT HEATER CORE
DO IT FOWARD AND BACK FLUSH / COVER DIST. CAP OR YOU WILL
FILL IT WITH WATER / IF IT CLEARS OUT HOOK HOSES BACK UP AND
BLEED AIR OUT OF SYSTEM / IF IT DOES'T CLEAR OUT HEATER CORE
WILL NEED TO BE REPLACED (DASH COMES OUT TO DO HEATER CORE)

If you look by the firewall there are two hoses. They go into the heater core. When the engine is warmed up, and the heater on. Feel if both hose are equally warm. Be carefull you don't burn yourself. A valve lets the hot water circulate through the heater core therefore providing heat. Also feel if the heater core is warm under you dash on the passenger side. If it feels cold then water is not circulating. Locate the valve, which is usually vacuum operated. See if there is vacuum there when the heater is in the on position.

either have a bad heater control valve, thermostat is not opening, or heater core is stopped up

Heater Core possibly leaking, good chance of air temperature sensor being disturbed. Check floor carpet under dash to see if its wet or sticky...sign of antifreeze! Good Luck!

check your radiator coolant, thats what I did and it blows warm again

With the car warmed up,feel the heater hoses,are they both warm/hot near the firewall in the engine compartment?If they are both warm/hot,then the heater core should have circulation through it and it may be a "blend door" issue in the heater a/c assembly.If they are not warm/hot then the heater core could be dirty and plugged up and needs to be flushed.
If the above does not help,a drain and vacuum of the cooling system and vacuum refill will remove any trapped air.
Sometimes,clamping off the top rad hose and holding the engine rpm at 2000 rpm will force trapped air out of the heater core.(Watch temp gauge when trying this)
Lastly,a NEW thermostat doesn't mean its working correctly.